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Recent Challenges in Actuarial Science ST09CH01_Embrechts ARjats.cls July 30, 2021 15:51 Annual Review of Statistics and Its Application Recent Challenges in Actuarial Science Paul Embrechts and Mario V. Wüthrich RiskLab, Department of Mathematics, ETH Zurich, Zurich, Switzerland, CH-8092; email: [email protected], [email protected] Annu. Rev. Stat. Appl. 2022. 9:1.1–1.22 Keywords The Annual Review of Statistics and Its Application is actuarial science, generalized linear models, life and non-life insurance, online at statistics.annualreviews.org neural networks, risk management, telematics data https://doi.org/10.1146/annurev-statistics-040120- 030244 Abstract Copyright © 2022 by Annual Reviews. For centuries, mathematicians and, later, statisticians, have found natural All rights reserved research and employment opportunities in the realm of insurance. By defini- tion, insurance offers financial cover against unforeseen events that involve an important component of randomness, and consequently, probability the- ory and mathematical statistics enter insurance modeling in a fundamental way. In recent years, a data deluge, coupled with ever-advancing information technology and the birth of data science, has revolutionized or is about to revolutionize most areas of actuarial science as well as insurance practice. We discuss parts of this evolution and, in the case of non-life insurance, show how a combination of classical tools from statistics, such as generalized lin- ear models and, e.g., neural networks contribute to better understanding and analysis of actuarial data. We further review areas of actuarial science where the cross fertilization between stochastics and insurance holds promise for both sides. Of course, the vastness of the field of insurance limits our choice of topics; we mainly focus on topics closer to our main areas of research. 1.1 ST09CH01_Embrechts ARjats.cls July 30, 2021 15:51 1. INTRODUCTION Early pioneers in insurance mathematics were the Dutch statesman Johan de Witt (1625–1672), with his essay ”The Worth of Life Annuities to Redemption Bonds,” and the Swiss theologian and mathematician Jakob Bernoulli (1655–1705), with his masterpiece Ars Conjectandi,where he proved an early version of the law of large numbers. In his correspondence with Gottfried Wilhelm Leibniz, Jakob Bernoulli mentioned in the context of his new asymptotic theory that the most important part of his work was still missing—namely, the application of his theoretical results to real world problems. Leibniz was not too enthusiastic about Bernoulli’s idea and argued that Bernoulli’s model was much too simple to answer real-world questions. It was his nephew Niklaus Bernoulli who later applied his uncle’s theory to mortality computations; we refer readers to Bolthausen & Wüthrich (2013). From the very beginning, actuarial science has been a discipline in statistical modeling that has been driven by practical problems in insurance. Teaching rigorous mathematical lectures on actuarial science only gained ground much later. The theoretical cornerstones of actuarial sci- ence were the work of Cramér (1930, 1994) and the book by Bühlmann (1970), who introduced the stochastic model approach toward non-life insurance. The latter was in contrast with the more deterministic view of life insurance at that time. In an influential 1989 editorial in the ASTIN Bul- letin, the editor then, Bühlmann (1989), famously introduced the so-called Actuary of the Third Kind, an actuary who uses his/her technical skills not only on the liability side of the insurance company’s balance sheet but also on the asset side. This actuary stands between the First Kind (the deterministic model–guided life actuary), the Second Kind (the stochastic model–oriented non-life actuary), and the Fourth Kind (the enterprise risk management–oriented actuary). These different kinds should not be interpreted as a reinvention of the actuarial profession; rather, they reflect the evolving societal conditions (demographic, technological, environmental, political, and legal) within which the actuarial profession fulfills its important tasks. On several occasions, we have defined the Actuary of the Fifth (final!) Kind as a data driven and model guided, criticaland socially responsible financial decision maker in an ever changing world governed by uncertainty. As such, the Fifth Kind is not all that distant from the etymology of the word actuarius, originating in the mid-sixteenth century as meaning copyist or account keeper. In Roman times, the actuar- ius was a quartermaster keeping the legion’s books—so, surely, someone strongly linked with and helpful in reaching business or strategic decisions based on data. Lester (2004) discusses the major challenges facing the actuarial profession going forward. They include the following: (a) the world has become a more uncertain place, (b)thereare numerous vigorous competitors, (c) the corporate governance and transparency push is placing increasing responsibility on boards and senior management, (d) communication is becoming a key success factor, and (e) the actuarial vocation is growing and spreading. Though written in 2004, points a–e not only remain true but have become more accentuated. Modern society no doubt offers many new challenges for actuaries. Examples include supply chain insurance, crop insurance, longevity bonds, the evolving world of catastrophe insurance, pandemic bonds, para- metric insurance, and innovative pension systems in a historically low-interest-rate environment, as well as the always-present market for insurance linked securities (ILSs). Areas like personalized medicine and telematics for auto insurance are newly born, drones take to the sky, and robots replace humans at an increasing rate. Several (but surely not all) of these new developments are driven by so-called big data and data science. The only viable way forward for actuaries is to embrace data science techniques or work closely with data scientists. A key anchor point, however, remains that an insurance product by (legal) definition offers a policyholder the relief of losses due to risks encountered. These products have to be well defined, technically understood, properly 1.2 Embrechts • Wüthrich ST09CH01_Embrechts ARjats.cls July 30, 2021 15:51 marketed, regulatory agreed on, to a certain extent not discriminating, and correctly priced and reserved, as well as clearly communicated. This will always call for actuarial understanding and an education that goes beyond mathematics and statistics. A modern actuarial qualification includes education in legislation, economics, accounting, professional behavior, and communication, and such a qualification also asks for a recognized program of continuous professional development to not lose sight of the evolving state of the art. In this review we mainly focus on recent challenges of statistical modeling in actuarial science. Tothis purpose, we divide actuarial science into three different branches: In Section 2, we discuss challenges in non-life insurance modeling; in Section 3, we present recent developments in sta- tistical modeling of life insurance; and in Section 4, we study reinsurance, risk management and specific applications to operation risk and cyber risk. This classification in three branches isquite typical—on the one hand, there are legal reasons for this partition of insurance because products in these three different branches typically require different legal entities for their marketing and sale. On the other hand, these branches have rather different characteristics (risk drivers) that re- quire different statistical modeling approaches. A bit outstanding is health and accident insurance, which has features of both life and non-life insurance as well as a strong intersection with social insurance, the latter being organized quite differently from the others. The confines of a relatively short article on a field as vast as actuarial science limit notonlythe topics we can treat but also the depth we can go into for those topics we discuss. We very much hope, however, that the more statistically oriented reader will get a good feeling for the kind of statistical techniques that enter the field of actuarial science. The rather extensive list of references offers sources for further reading. 2. NON-LIFE INSURANCE MODELING 2.1. A Brief Overview of Non-life Insurance The term non-life insurance is mainly used in Europe, and it summarizes all direct insurance products that are different from life insurance. In the United Kingdom, non-life insurance is also termed general insurance, and in the United States and Canada it is called property and casualty insurance. In non-life insurance, one typically distinguishes two functions: There is the pricing actuary who designs and prices (new) insurance products, and there is the reserving actuary who predicts cash flows of insurance claims (which typically run over multiple years). These predictions are used for insurance accounting, risk management, and product development. 2.2. Non-life Insurance Pricing Non-life insurance pricing is the actuarial domain that runs at the forefront of statistical modeling and data science. It is a traditional discipline where the statistical modeling cycle is explored; here we refer readers to McCullagh & Nelder (1983, section 2.1) and Box & Jenkins (1976). Typically in non-life insurance, one faces the problem of having a heterogeneous portfolio of insurance policyholders, and one aims at charging risk-adjusted prices to each of these customers.
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